**2. Results**

As the underlying model for phase transitions with a conserved order parameter we consider a Cahn–Hilliard equation

$$
\partial\_t \phi = \gamma \Delta \mu, \qquad \mu = -\epsilon^2 \Delta \phi + B'(\phi); \tag{1}
$$

see Section 4 for details. First, we consider coarsening in a rectangular cuboid using standard boundary conditions **n** · ∇*φ* = **n** · ∇*μ* = 0 on *∂*Ω. Figure 1 shows snapshots of the results within the large-time regime, visualized in various ways. The interface area is minimized, and the structure thus coarsens in time. To analyze this in a statistical manner requires either a larger domain or more samples. Our idea is to combine both by using samples which are distinct from each other but fit together to form a large and extendable structure. The boundary conditions in the current setting enforce the level lines of the interface to be perpendicular to *∂*Ω. They in addition do not fit to each other and thus do not allow combination of different samples. To overcome these limitations, we consider a smaller domain, again a rectangular cuboid and the boundary conditions introduced in Section 4 which specify the values of *φ* and the normal flux ∇*φ* · **n** such that opposite sides match. The first approach only has two distinct boundaries, *N* = 2. Figure 2 shows four samples, all obtained with different initial conditions and considered at the same time instance. The structures fit together and any translation in x- or y-direction by the width of the domain will also fit. The individual figures are provided in SI as Figures S1–S4; print them and try it out. The figures are part of an art project, *M* = 100 individual samples have been computed and printed on Alu-Dibond in size 20 cm × 20 cm, creating a 200 cm × 200 cm figure which can be displayed in 100! ≈ 9.332622 · 10<sup>157</sup> variations.

**Figure 1.** Typical structure within the large-time regime, visualized as *φ* = 1 in Ω, *φ* = 0.5 at *z* = 0.01, 0.13, 0.25, 0.37, 0.49 and *φ* = 0.5 at *z* = 0.01, 0.13, 0.25, 0.37, 0.49 projected to *z* = 0, from left to right. The boundary conditions are **n** · ∇*φ* = **n** · ∇*μ* = 0 on *∂*Ω. The corresponding videos of the coarsening process are provided in SI as Videos S5–S7.

**Figure 2.** Four samples with identical boundaries but distinct inner structure. The samples are translational invariant in *x*- and *y*-direction.

Even if the inner structure is unique for each tile, the boundaries in *x*- and *y*-direction are the same and recurrences are visible. To improve on this issue, we consider a second approach, which is less flexible in terms of arrangemen<sup>t</sup> of samples but minimizes possible recurrences. As a compromise of computational cost and visual impression we consider tilings with *N* = 10 different boundary conditions. This improves the impression of a tiling with no recurrence since systematic recurrence in a row, a column, or diagonally can be avoided by careful assembly of the tiles. Repetitions do not only appear less frequently but also in a pattern that is much less obvious. To see this recurrence without knowing the construction process and explicitly searching for them is almost impossible. We consider two different internal realizations each, leading to *M* = 10. Within the proposed pattern determined by the boundary conditions the inner realization are randomly chosen to construct an infinite tiling, where recurrences are almost invisible.

A software is developed to visualize the infinite structure. We consider visualizations with five projected level lines of the interface. The software allows zooming in and out, evolve forward and backward in time as well as traverse the infinite tiling horizontally and vertically. Figure 3 shows some screenshots, starting from an early time instant and a low zoom factor (a), going to a late-time instant of this setting (b), zooming into the structure (c), evolving along a trajectory in space and time, which keeps the interface area constant (d), and going back to the initial state (a). Videos of the journey through space and time are provided in SI as Videos S8 and S9.

**Figure 3.** Screenshots of the visualization software, here in addition color-coded according to the individual tiles used. The dark magenta lines indicate the user-interaction. Moving the mouse horizontally evolves time, moving it vertically zooms in and out.

As the proposed approach is in principle not restricted to rectangular cuboidal domains various possibilities for applications can be imagined. Besides wallpaper design, they range from fashion design with individualized clothes to camouflage patterns of automotive prototypes. Here, we highlight a more entertaining application, a Rubik's cube which always fits, but has 24 different fields; see Figure 4.

**Figure 4.** A Rubik's cube which always fits even if all tiles are different. A video is provided in SI as Video S10.
